1.3 Black hole perturbation theory
1.3.3 Quasinormal mode spectrum of Kerr black holes and its geometric interpreta-
Chapter 9 is a research paper published as:
• Huan Yang, David A. Nichols, Fan Zhang, Aaron Zimmerman, Zhongyang Zhang, and Yan- bei Chen, Quasinormal-mode spectrum of Kerr black holes and its geometric interpretation, Physical Review D86104006 (2012).
1.3.3.1 Motivation and significance
Initially Chen found numerically that for certain Kerr black hole spins there are degenerate frequen- cies among different sets of quasinormal modes (QNMs). In order to explain this degeneracy, an
analytical understanding for Kerr QNM frequency is needed. We worked out the QNM frequency formula in the eikonal limitl1 using WKB techniques, which agrees with numerical results very well. Using the analytical formula, the mode degeneracies are naturally explained and moreover, we showed these Kerr QNMs are intimately related to null particles moving in spherical orbits of Kerr. In this work we have built up analytical tools which are not only applicable for analyzing Kerr QNMs but has also been proved to be useful for the work in later Chapters 10–11.
1.3.3.2 Summary of main results
Quasinormal modes (QNMs) of black-hole spacetimes are the characteristic modes of linear per- turbations of black holes that satisfy an outgoing boundary condition at infinity and an ingoing boundary condition at the horizon. For generic Kerr black holes, they are well understood and can be calculated quite accurately, for example using the numerical algorithm developed by Leaver [97].
In addition, for modes withl1, there is a well-known, intuitive geometric correspondence between high-frequency quasinormal modes of slowly-rotating Kerr black holes and null geodesics that reside on the light-ring (often called spherical photon orbits) [98]:
ωlmn≈L 1
√27M +m 2a
27M2 −iN 1
√27M (1.31)
whereL≡l+ 1/2 andN=n+ 1/2. The real part of the mode’s frequency relates to the Keplerian orbital frequency for the spherical photon orbit 1/(√
27M), and the Lense-Thiring-precession fre- quency of the orbit 2S/(3M)3 = 2a/(27M2) (S is the angular momentum of the Kerr black hole);
the imaginary part of the frequency corresponds to the Lyapunov exponent of the orbit 1/(√ 27M).
As a result, we may expect closed photon orbits to play an important role in the structure of a spacetime’s QNM. It is however nontrivial to generalize this geometric correspondence to generic Kerr black holes. The main difficulty comes from the fact that radial and angular Teukolsky equa- tion both contain two constants to be determined by the boundary conditions — angular and radial eigenvalues, and hence these two second-order differential equations have to be solved jointly.
Instead of directly solving the joint second-order differential equations, we try to derive the algebraic relations between the two eigenvalues based on the two Teukolsky equations. The first relation was worked out by Iyer and Will [99]. They used the fact that the radial Teukolsky equation describes a scattering problem, and applied WKB techniques to showV(ωR, r0) =V0(ωR, r0) = 0 and:
ωI = (n+ 1/2)
p2d2Vr/dr2∗
∂Vr/∂ω r
0,ωR
(1.32)
wherer0is the peak position andωlmn=ωR−iωI(the angular eigenvalueAlmis implicitly contained
in the expression for Vr). The second relation was discovered by us, by noticing that the angular Teukolsky equation can be transformed into a boundary state problem, which is extensively studied in quantum mechanics. We apply a WKB approximation on this bound-state problem, and obtain the second relation between angular eigenvalueAlm and QNM frequency ωR
Z θ+ θ−
dθ r
a2ωR2cos2θ− m2
sin2θ+ARlm = (L− |m|)π , (1.33) where the integral is performed in the classical regime (integrand≥0). Comparing our WKB calcu- lation to the leading-order, geometric-optics approximation to scalar-wave propagation in the Kerr spacetime, we then draw a correspondence between the real parts of the parameters of a quasinor- mal mode and the conserved quantities of spherical photon orbits. At next-to-leading order in this comparison, we relate the imaginary parts of the quasinormal-mode parameters to the Lyapunov exponent of the scalar wave. In particular, the QNM frequency is
ωlmn=Lωorb+mωprec−iN γ+O(1/L) (1.34) where ωorb is the orbital frequency of the corresponding photon orbit, ωprec is the corresponding precession frequency andγis the Lyapunov component of the orbit.
The above QNM formula is checked against numerical scalar QNM values and the relative errors are shown to be under 0.06/L2. Moreover, it manifests the geometric correspondence between Kerr QNMs and the null rays propagating in spherical orbits of Kerr spacetime. With the correspondence, we also make other observations about features of the QNM spectrum of Kerr black holes that have simple geometric interpretations. First, we find that for near extremal Kerr black holes with a/M → 1, a significant fraction of the QNMs have their real frequencies approach m times the angular frequency of the horizon and a decay rate that rapidly falls to zero; we explain this in terms of a large number of spherical photon orbits that collect on the horizon for extremal Kerr holes.
This phenomenon will be further investigated in Chapter 10. Second, we use the geometric-optics interpretation given by Eq. (9.3) to explain a degeneracy in the QNM spectrum of Kerr black holes, in the eikonal limit, which also manifests itself, approximately, for smalll. The degeneracy occurs when the orbital and precession frequencies, ωorb andωprec are rationally related (i.e., ωorb/ωprec =p/q for integerspandq) for a hole of a specific spin parameter, and when the corresponding spherical photon orbits close. By substituting this result into Eq. (9.3), one can easily see that modes withl andm become degenerate with those of indexesl0 =l+kq andm0 =m−kpfor any non-negative integerk, in the eikonal limit.
1.3.3.3 My specific contributions
I performed most of the analytical calculations in this work, and wrote the initial draft of the paper, which was dramatically improved by all the authors.